can you give me few hints how to solve this problem ?
Relation R on the set P(A)
A = {a,b,c,d} is a set of four elements. We also have relation R on the set P(A), which is defined R={(A,B)│A ⊆ B. The difference between count of elements of A and count of elements of B is even number.
Is relation R partial order on the set P(A) ? Give reasons for your answer. If the answer is yes, draw Hass diagram.
Well, 0 is even so $(A,A)\in R$ - that's reflexivity. If $(A,B)$ and $(B,A)$ are in $R$, then $A\subseteq B\subseteq A$ implies $A = B$ - that's antisymmetry. And if $(A,B)$ and $(B,C)$ are in $R$, then $A\subseteq B\subseteq C$ implies $A\subseteq C$. Now, $|C|-|A| = |C|-|B|+|B|-|A|$ is a sum of two even numbers, hence even, so $(A,C) \in R$ - that's transitivity.